PRINGLE PATTISON Y EL VALOR DE LA PERSONA HUMANA

The distribution of available volumes at the best can be fitted by a gamma distribution with an exponent less than unity, meaning that the most probable value of the volume is much smaller than the average value. Both the value

of the spread S and the quantity available at the bid and the ask, Φb, Φa, give

information on the willingness of liquidity providers to enter a trade. One would like to understand the relation between these quantities – intuitively, large spreads are more favorable to liquidity providers and should attract larger volumes. More generally, it would be extremely interesting to have a theory for the shape of the

ORDER BOOK DYNAMICS 83 whole order book, i.e. the relation between the available volume and the distance from the best price.

The approach of Glosten and Sandas attempts to answer the above questions, within a framework where market orders are informed trades (Glosten [1994], Sandas [2001]). The idea is now that information is time dependent and modelled as a random variable that gives the predicted future variation of the midpoint,

which we call (in conformity with the above notation) nr(n, n + `). Just before

the nth trade, a liquidity provider considers the volume of the queue at the ask,

Φa,n and decides to add an extra (infinitesimal) limit order if its expected gain,

conditional on execution, is greater than some minimum value Gmin ≥ 0. This

reads:

E[mn+`− mn|n = 1, vn ≥ Φa,n] ≤

Sn

2 − Gmin. (9.2)

At this stage, Glosten and Sandas add several questionable assumptions. A cru- cial one is that the volume that the informed trader chooses to trade is propor-

tional to the information he has: vn= αr(n, n + `), independently of the shape of

the book at that moment, and in particular of the available volume at the ask. He is prepared to walk up the book if necessary, which occurs with only a very small probability in practice: as discussed in Section 6.1, trading is, in fact, discre-

tionary. Introducing the probability of information content P`(r), and dropping

the index n for convenience, the above conditional expectation inequality reads:

Z +∞ Φa/α rP`(r)dr ≤  S 2 − Gmin  Z +∞ Φa/α P`(r)dr, (9.3)

where we have used the fact that information is assumed to be reliable, i.e. the expected mid-point change is indeed given by the informed trader prediction. In

order to achieve a quantitative prediction, Sandas further assumes that P`(r) has

an exponential shape:27 P`(r) = βe−βr −→ Φa α + 1 β ≤ S 2 + Gmin. (9.4)

In fact, this calculation can be reinterpreted to give the total volume of orders

available Φ< at a price less or equal to p = m + S/2, and therefore makes a

prediction for the shape of the order book:

Φ<(p) = α(p − m) − αGmin−

α

β, (9.5)

i.e. a linear order book with slope α and, in principle, a negative intercept. (The prediction for the buy side of the book is obvious by symmetry). Note that within

27_{This exponential assumption is in fact not so important. For example, a pure power-law} distribution P`(r) ∝ r−1−µ when r > r0 would lead to the following result instead: Φa/α ≤ (1 − µ−1)[S/2 + Gmin] (µ > 1).

ORDER BOOK DYNAMICS 84 this framework, the volume dependent impact of market orders is by assumption

linear: R`(v) = v/α, which we already know is quite a bad representation of real

data, where impact is always strongly sublinear (see Section 5.1). Altogether, this model fares quite badly when compared with empirical data:

• The order book intercept, which should be negative according to the model, is found to be positive when the model is fitted to empirical data. suggesting negative costs for placing limit orders;

• The slope α, when obtained from the slope of the order book, is found to be ten times larger than when obtained from direct impact estimates.

• As mentioned above, the empirical shape of order books is non-monotonic, exhibiting a maximum away from the best price. This is not accounted for by the model.

The reason for such a failure is essentially that, as discussed in Section 6.1, as shown by Farmer et al. [2004], the volume of the incoming market order is in fact strongly correlated with the available volume at the best price. This is in fact why impact is sublinear in volume, and is at the heart of the liquidity game we have been detailing in the previous pages. One cannot consider that the market order flow is an exogenous process to which the limit order flow must adapt – rather, the two coevolve in a strongly intertwined manner.

One can however directly test Eq. (9.2) on empirical data, without any further theoretical assumptions, much as we did in the previous section. We choose ` = 1, 10, 100 and identify a “neutral line” in the S, Φ plane separating the region (above that line) where executed limit orders are profitable from a region where

they are costly (see Fig. (21), and Eisler et al. [2008]). One sees that after

the ` = 1 trade the separation line is flat and is located around the value of the average spread. This means that the value of the spread is such that limit orders and markets order break even on average at high frequencies, as discussed

in section 7.3. However, judged on longer time scales, the profitability of a

limit order behind a large preexisting order only becomes positive for spreads significantly larger than the average. In other words, correlations between spread and volume, of the type predicted by the Glosten-Sandas model (Eq.9.2) indeed appear on longer time scales.

9.3 Statistical models of order flow and order books